In truth, it's not the sleeping position that introduces lower back pain–– it's the posture.
Keep your spine relaxed
Support pillows can help
If you sleep on your back, you don't have the problem of straightening your spine, instead, your lower back pressure comes from the inability to keep your spine in a natural curve. The spine curve near your lower back is experiencing a pull from your hip, thus forcing it to straighten unnaturally, causing it to tighten up and become over-stressed.
Use a half-moon bolster underneath your knees. This will raise your legs slightly and tilt your hip forward to relieve its pull on your lower back.
If you sleep on the side, you need to keep your spine straight and aligned with your hip. Your shoulder and your hips are the pillars of your body weight. They serve as the two ends of a rope bridge, if either one is crooked, the center of the bridge will sag and experience all the pressure and weight, that's your lower back. You need to make sure both your shoulder, and hip are upright, so your body weight is evenly distributed without concentration near the lower back.
3. Stomach Sleepers
If you sleep on your stomach, your spine is probably being forced to straighten unnaturally against its curve, causing pressure from your shoulders to your lower back.
Sleep on a thin, low profile pillow to keep your neck flat against your upper back. Additionally, insert a small flat pillow near your lower stomach to allow your spine to maintain its natural curve near the tailbone to relieve the pull on your back.